AP Calc Flashcard Quiz
Section 2.5 o Give the four scenarios in which the graph of f(x) is not differentiable at x = a with explanation as to why they are not differentiable
(1) f(x) is discontinuous at x = a Explanation: the graph must be continuous at the point in order for it to be differentiable at the point (2) f(x) contains a vertical tangent line at x = a Explanation: VTL's have undefined slopes and thus not differentiable (3) f(x) contains a cusp at x = a Explanation: The tangent lines become vertical as you get infinitely close to the point (4) f(x) contains a corner at x = a Explanation: The derivative values aren't matching on both sides of the corner
Section 5.1 o Evaluate the following basic anti-derivatives (1) ∫ x^n dx (2) ∫ k dx ; where k is a constant (3) ∫ e^x dx (4) ∫ 1/x dx (5) ∫ Sin(x) dx (6) ∫ Cos(x) dx (7) ∫ Sec^2(x) dx (8) ∫ Csc^2(x) dx (9) ∫ Sec(x)Tan(x) dx (10) ∫ Csc(x)Cot(x) dx
(1) x^(n+1)/(n+1) + C (2) kx + C (3) e^x + C (4) ln|x| + C (5) −Cos(x) + C (6) Sin(x) + C (7) Tan(x) + C (8) −Cot(x) + C (9) Sec(x) + C (10) −Csc(x) + C
Pre Calc Review o The side opposite 30 degrees is always.... o The side opposite 45 degrees is always... o The side opposite 60 degrees is always...
1/2 √2/2 √3/2
Section 4.6 o Fill out the blank spaces in the table below Graph of f(x) | Graph of f′(x) | Graph of f′′(x) Local Max/Local Mins →__________________→_______________ Inflection Points →_________________→________________
Graph of f(x) | Graph of f′(x) | Graph of f′′(x) Local Max/Local Mins → x-int →_______________ Inflection Points → Local Min/Max → x-int
Algebra Review Graphs o Make a sketch of the following parent graphs with the key points on the graph: ▪ f(x) = e^x ▪ f(x) = ln (x)
Search this stuff up if you're not sure, I can't upload graphs on Quizlet
Algebra Review Graphs o Make a sketch of the following parent graphs with the key points on the graph: ▪ f(x) = x^2 ▪ f(x) = −x^2 ▪ f(x) = x^3 ▪ f(x) = −x^3
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Algebra Review Graphs o Make a sketch of the following parent graphs with the key points on the graph: ▪ f(x) = √x ▪ f(x) = x^(1/3)
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Trig Review Graphs o Make a sketch of the following parent graphs with the key points on the graph: ▪ f(x) = Sin(x) ; f(x) = Cos(x) ; f(x) = Tan(x)
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Pre Calc Review o Express Tan(θ) and Cot(θ) in terms of Sin(θ) and Cos(θ)
Tan(θ) = Sin(θ)/Cos(θ) Cot(θ) = Cos(θ)/Sin(θ)
Section 1.8 Part 1 o Explain what limx→+∞ f(x) = ∞/∞ is telling us and what we still need to figure out
The numerator and denominator are both going towards infinity but we still need to determine which one is getting there faster
Section 3.1 o Give the derivatives for all six trig functions
f(x) = Sin(x) f′(x) = Cos(x) f(x) = Cos(x) f′(x) = −Sin(x) f(x) = Tan(x) f′(x) = Sec^2(x) f(x) = Csc(x) f′(x) = [−Csc(x)][Cot(x)] f(x) = Sec(x) f′(x) = [Sec(x)][Tan(x)] f(x) = Cot(x) f′(x) = −Csc^2(x)
Section 6.3 o What are the two different versions of an accumulation function and how do you take their derivatives?
o (1) F(x) = ∫ (g(x) top, a bottom) f(t) dt; where a is a constant F′(x) = f(g(x)) ∙ g′(x) o (2) F(x) = ∫ (g(x) top, h(x) bottom) f(t)dt F′(x) = f(g(x)) ∙ g′(x) − f(h(x)) ∙ h′(x)
Section 4.2 o Steps to find the Global Min / Max on an interval [a, b]
o (1) Find all local min and max on the interval o (2) Find the y-values at the end points of the interval o (3) Global Min = smallest number ; Global Max = largest number
Section 4.3 o What are the steps to find the intervals of concavity of a function?
o (1) Find all values of c such that f′′(c) = 0 o (2) Test the intervals on #-line checking if f′′(c) is Pos or Neg o (3) Concave Up for all positive regions ; Concave Down for all negative regions
Section 4.1 o What are the steps to find the intervals of increase / decrease of a function?
o (1) Find critical values and put them on a number line o (2) Test the intervals on #-line checking if f′(x) is Pos or Neg o (3) Increasing for all positive regions ; Decreasing for all negative regions
Pre Calc Review o Sin(θ) is positive in which two quadrants? o Cos(θ) is positive in which two quadrants? o Tan(θ) is positive in which two quadrants?
o 1 & 2 o 1 & 4 o 1 & 3
Pre Calc Review o Sin(θ) is negative in which two quadrants? o Cos(θ) is negative in which two quadrants? o Tan(θ) is negative in which two quadrants?
o 3 & 4 o 2 & 3 o 2 & 4
Section 6.2 o What is a separable differential equation and how do you solve it?
o A differential equation written in terms of two different variables How to solve: Step 1: Separate the variables on the two sides of the equation, bringing all y's and dy to one side and all x's and dx to the other Step 2: Integrate both sides and isolate y by itself by whatever algebra process is necessary to do so.
Section 1.11 o Give the formal definition of continuity at a single point
o A function is continuous at x = a if: f(a) is defined on the graph limx → a− f(x) = limx → a+ f(x) = f(a)
Section 6.1 o What is a differential equation?
o An equation representing a derivative (rate of change) typically expressed in the form dy/dx=
Section 5.3 o What is a "definite" integral and what does it represent?
o An integral with "bounds" on top and bottom: ∫ (a bottom, b top) f(x)dx o It represents the area under the curve on the interval [a, b]
Pre Calc Review o Where is Sin(θ) = −1 on the unit circle? o Where is Cos(θ) = −1 on the unit circle?
o Any angle that lands on the negative side of the y-axis o Any angle that lands on the negative side of the x-axis
Pre Calc Review o Where is Sin(θ) = 1 on the unit circle? o Where is Cos(θ) = 1 on the unit circle?
o Any angle that lands on the positive side of the y-axis o Any angle that lands on the positive side of the x-axis
Pre Calc Review o Where is Sin(θ) = 0 on the unit circle? o Where is Cos(θ) = 0 on the unit circle?
o Any angle that lands on the x-axis o Any angle that lands on the y-axis
Pre Calc Review o Where is Tan(θ) = 0 on the unit circle? o Where is Tan(θ) = undefined on the unit circle?
o Any angle that lands on the x-axis (Since Sin = 0 at same location) o Any angle that lands on the y-axis (Since Cos = 0 at same location)
Pre Calc Review o Where is Cot(θ) = 0 on the unit circle? o Where is Cot(θ) = undefined on the unit circle?
o Any angle that lands on the y-axis (Since Cos = 0 at same location) o Any angle that lands on the x-axis (Since Sin = 0 at same location)
Section 5.8 o Integral Set Up for Area Between Two Curves with Respect to x
o Area = ∫(x2 top, x1 bottom) [Top Function] − [Bottom Function] dx where x1 & x2 are the smallest and largest x values on the bounded region
Section 5.10 o Integral Set Up for Area Between Two Curves with Respect to y
o Area = ∫(y2 top, y1 bottom) [Right Side Function] − [Left Side Function] dy where y1 & y2 are the smallest and largest y values on the bounded region
Section 2.1 o How do you calculate Average Rate of Change of f(x) on [a,b]?
o Avg ROC = [f(b) − f(a)] / (b-a)
Section 4.7 o Formulas for Average Velocity and Average Acceleration
o Avg Velocity = (s(b) − s(a)) / (b − a) where s(t) = position o Avg Acceleration = (v(b) − v(a)) / (b − a) where v(t) = velocity
Section 1.8 Part 2 o Give the three cases for Horizontal Asymptotes on the graphs of rational functions
o Case 1: Degree of Num < Degree of Den - H.A. is always 0 o Case 2: Degree of Num = Degree of Den - H.A. is division of Leading Coefficients o Case 3: Degree of Num > Degree of Den - There is not H.A. on the graph (and therefore the limit value is either +∞ or − ∞)
Section 4.1 o How can you use the graph of f′(x) to determine where f(x) is increasing or decreasing?
o Check the intervals on which the given graph is above or below the x-axis (checking where f′(x) is positive or negative)
Section 4.3 o How can you use the graph of f′(x) to determine where f(x) is concave up or down?
o Check the intervals on which the given graph is increasing or decreasing
Section 5.3 o Give the integral formulas for displacement and total distance
o Displacement = ∫ (a bottom, b top) v(t) dt o Total Distance = ∫ (a bottom, b top) |v(t)| dt
Section 4.7 o What is difference between distance and displacement?
o Distance = total units traveled over a time interval o Displacement = ending location in relation to starting position over a time interval
Section 1.3 o Explain how to find the limit of a composite function
o Given a composite function f(g(x)) then: limx→a f(g(x)) = f(limx→a g(x))
Section 5.7 o Explain how to calculate Average Value of a Function on [a, b]
o Given a continuous function f(x) on [a,b] then then: Average Value of f(x) on [a, b] = 1/(b − a) ∙ ∫(a bottom, b top) f(x) dx
Section 5.5 o Explain the Fundamental Theorem of Calculus
o Given a continuous function f(x) on [a,b] then: Area Under f(x) on [a, b] = ∫ (a bottom, b top) f(x) dx = F(b) − F(a) where f(x) = F′(x)
Section 1.10 o Explain Removable vs Non-Removable Discontinuity in terms of crossing out factors
o If a factor (x − k) can be crossed out in the numerator and denominator of a given function then this means there is a single hole on the graph at x = k and it is labeled as removable discontinuity o If a factor (x − k) remains in the denominator of a given function after it is fully simplified then this means there is a vertical asymptote on the graph at x = k and it is labeled as non-removable discontinuity
Section 1.12 o Give an explanation of the Intermediate Value Theorem o Draw a picture that supports the explanation
o If a function f(x) is continuous on a given interval [a, b] then there must exist some value of c on the same interval such that f(c) = M where f(a) < M < f(b)
Section 1.6 o Explain the Squeeze Theorem
o If f(x) < g(x) < h(x) and it is known that both: limx→a f(x) = L and also limx→a h(x) = L Then the Squeeze Theorem says that limx→a g(x) = L
Section 3.11 o Explain the Mean Value Theorem
o If f(x) is a continuous function on [a, b] and also differentiable on (a, b) then there must exist some value of c where a < c < b such that the Average ROC on the interval [a, b] is equal to the Instantaneous ROC at x = c f′(c) = Avg ROC on [a, b] for some value of c
Section 4.4 o Explain the 2nd Derivative Test for Identifying Relative Extrema
o If f′(c) = 0 & f′′(c) > 0 [Concave Up] then x = c is a Local Min o If f′(c) = 0 & f′′(c) < 0 [Concave Down] then x = c is a Local Max
Section 3.8 o Give the formula for the derivative of an inverse function
o If g(x) = f−1(x) then g′(x) = 1 / f′[f−1(x)]
Section 3.4 o Explain the Chain Rule for derivatives
o If h(x) = f(g(x)) Then h′(x) = f′(g(x)) ∙ g′(x)
Section 3.3 o Explain the Quotient Rule for derivatives
o If h(x) = f(x) / g(x) Then h′(x) = [g(x) ∙ f′(x) − f(x) ∙ g′(x)] / g(x)^2
Section 3.3 o Explain the Product Rule for derivatives
o If h(x) = f(x) ∙ g(x) Then h′(x) = f(x) ∙ g′(x) + g(x) ∙ f′(x)
Section 3.7 o Explain L'Hopsitals Rule
o If limx→a f(x)/g(x) = 0/0 or ∞/∞ then the limit is equal to f′(a)/g′(a)
Section 4.3 o What are inflection points?
o Locations where a function changes concavity
Section 4.7 o When is a particle moving to the left? Moving to the right? At rest?
o Moving to the left when velocity is negative o Moving to the right when velocity is positive o At rest when the velocity is zero
Section 4.1 o A Local Min occurs when f′(x) switches from _________________ o A Local Max occurs when f′(x) switches from _________________
o Negative to Positive o Positive to Negative
Pre Calc Review o Give the one trig identity for Sin(2x) and the three trig identities for Cos(2x)
o Sin(2x) = 2Sin(x)Cos(x) o Cos(2x) = { Cos^2(x) − Sin^2(x) 1 − 2Sin^2(x) 2Cos^2(x) − 1
Pre Calc Review o Give the three Pythagorean Trig Identities
o Sin^2(θ) + Cos^2(θ) = 1 o 1 + Cot^2(θ) = Csc^2(θ) o Tan^2(θ) + 1 = Sec^2(θ)
Section 2.2 o Give two other synonyms for derivative of a function
o Slope of the Tangent Line o Instantaneous Rate of Change
Section 4.7 o What is definition of speed?
o Speed = |v(t)|
Section 4.8 o When is a particle speeding up? When is a particle slowing down?
o Speeding Up when v(t) and a(t) are matching (PP or NN) o Slowing Down when v(t) and a(t) are not matching (PN or NP)
Section 3.10 o Explain the steps of Implicit Differentiation
o Step (1): Take the derivative of both sides of the equation using derivative rules and multiply by dy/dx when taking the derivative of any term involving a y o Step (2): Solve this equation for dy/dx
Section 1.8 Part 2 o If f(x) is a rational function, what is limx → ±∞f(x) asking about?
o The HORIZONTAL ASYMPTOTE of the graph (if one exists)
Section 4.1 o What does it mean if f′(x) > 0 on a given interval [a,b]? o What does it mean if f′(x) < 0 on a given interval [a,b]?
o The derivative is positive and therefore the graph of f(x) is increasing (going up) on the same interval o The derivative is negative and therefore the graph of f(x) is decreasing (coming down) on the same interval
Section 4.9 o When you are finding the maximum or minimum value of a given function, what do you have to set equal to 0?
o The derivative of the function you are trying to maximize or minimize.
Section 6.6 o Explain how you know when to use the disc method compared to the washer method for volume
o The disc method is used if there is no physical separation between the shaded region and the axis you are rotating around. If there is any amount of physical separation, then you use the washer method
Section 1.10 o Explain Non-Removable Discontinuity and why it is called this
o The discontinuity is not able to be removed at x = a because there is either a vertical asymptote of a jump in the y-values on the graph
Section 1.9 o Explain what limx→a f(x) = #/0 is telling us about the graph
o The function is not defined at x = a because there is a VERTICAL ASYMPTOTE on the graph o This means the limits are going to be some version of "infinity" on either side of the asymptote
Section 6.7 o What is the solution to a given slope field? How do you draw the solution?
o The general function that is being displayed by the slope field o The graph of the function can be sketched if you were to label a particular point and then use the derivatives to follow the path of the point as it moves through the slope field. Like a leaf being pushed by a current.
Section 5.4 Part 2 o Explain trapezoidal area approximations
o The height of the trapezoid is the length of the sub-interval and the two bases of the trapezoid are the function values for the numbers on the endpoints of the sub-interval
Sections 1.1 & 1.2 o A limit value can only exist as x is approaching "a" if _____________
o The limit from the left as x approaches "a" matches the limit from the right as x approaches "a"
Section 6.1 o What is the solution to a differential equation?
o The original function where the derivative dy/dx came from
Section 4.1 o What are the critical values of a function?
o The values that make f′(x) = 0 or undefined
Section 5.4 Part 1 o Explain rectangular area approximations using left end points
o The width of the rectangle is the length of the sub-interval and the height of the rectangle is the function value of the number on the left hand side of the sub-interval
Section 5.4 Part 1 o Explain rectangular area approximations using right end points
o The width of the rectangle is the length of the sub-interval and the height of the rectangle is the function value of the number on the right hand side of the sub-interval
Section 5.4 Part 1 o Explain rectangular area approximations using midpoints
o The width of the rectangle is the length of the sub-interval and the height of the rectangle is the function value of the number that is the midpoint of the sub-interval
Section 1.5 o What does a limit value of 0/0 indicate about the graph of the function?
o There is a hole in the graph at one point so the function value doesn't exist but the limit value does
Section 2.4 o What is the relationship between the Tangent Line and the Normal Line?
o They are perpendicular to each other and thus have opposite reciprocal slopes
Section 1.10 o Explain Removable Discontinuity and why it is called this
o This means there is a single hole on the graph at x = a and therefore the function value is undefined for f(a) but the limit value must be a real number that matches on either side o It is called this because you can remove the discontinuity by re-writing the function as a piecewise function that also defines the function for the one single point and therefore it plugs up the hole
Sections 1.1 & 1.2 o What is limx→a f(x) asking?
o To give the y-value that the function f(x) is approaching as the value of x is getting infinitely close to "a"
Section 6.5 o Integral formula for volume created by using the Disc Method to rotate area around a given axis
o V = ∫(a bottom, b top) π ∙ r^2 dx where r is the distance from the function to the given axis
Section 6.6 o Integral formula for volume created by using the Washer Method to rotate area around a given axis
o V = ∫(b top, a bottom) πR^2 − πr^2 dx where R is the distance from the axis to the farther away function and r is the distance from the axis to the closer function
Section 5.2 o When do you use U-Substitution method of integration?
o When taking the anti-derivative of a composite function
Section 3.10 o In what situation would you need to use Implicit Differentiation?
o When you cannot (or don't want to!) explicitly solve for y within a given equation
Section 2.3 o What is a Horizontal Tangent Line code for in calculus?
o f′(x) = 0 [the slope of any horizontal line is 0!]
Section 2.2 o Give the limit definition for the derivative of f(x)
o f′(x) = limh→0 [f(x + h) − f(x)]/h
Section 2.2 o Give the three different notations for derivative of f(x)
o f′(x); dy/dx; d/dx
Section 1.7 o Give the two Special Trig Limits and their variations
o limx→0 Sin(x)/x = 1 or limx→0 x/Sin(x) = 1 o limx→0 (Cos(x) − 1)/x = 0 or limx→0 (1 − Cos(x))/x = 0
Section 4.7 o What is the derivative relationship between position, velocity and acceleration functions?
o v(t) = s′(t) o a(t) = v′(t) = s′′(t)
o Explain how |x − a| changes depending on the values of x and a
o |x − a| = { x − a if x − a > 0 −(x − a) if x − a < 0 or { x − a if x > a −(x − a) if x < a
Section 5.4 Part 3 o What is the conversion between the definite integral and a Reimann Sum?
o ∫ (a bottom, b top) f(x) dx = limn→∞ ∑ (n top, k=1 bottom) f(a + k ∙ ∆x) ∙ ∆x where ∆x = (b − a)/n
Section 3.9 o Give the derivatives of arccsc, arcsec(x) , arccot(x)
od/dx[arccsc(x)] = −1 / (|x|√x^2 − 1) od/dx[arcsec(x)] = 1 / |x|√x^2 − 1 od/dx[arccot(x)] = −1 / (1 + x^2)
Section 3.9 o Give the derivatives of arcsin(x) , arccos(x) , arctan(x)
od/dx[arcsin(x)] = 1 /√(1 − x^2) od/dx[arccos(x)] = −1 /√(1 − x^2) od/dx[arctan(x)] = 1 / (1 + x^2)
Section 6.4 o Integral formula for volume created by taking the following geometric shaped cross sections of a given area perpendicular to the x-axis ▪ Square ▪ Rectangular ▪ Semi-Circular ▪ Equilateral Triangle ▪ Isosceles Right Triangle
where s = f(x) − g(x) o Square V = ∫(a bottom, b top) s^2 dx o Rectangular V = ∫(a bottom, b top) s ∙ (height) dx o Semi Circular V = ∫(a bottom, b top) π/8 ∙ s^2 dx o Equilateral Trangle V = ∫(a bottom, b top) √3/4 ∙ s^2 dx o Isosceles Right Trangle V = ∫(a bottom, b top) 1/2 ∙ s^2 dx
